# On the Exact Inverse and the pth Order Inverse of Certain

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ON THE EXACT INVERSE AND THE pTH ORDER INVERSE OF CERTAIN
NONLINEAR SYSTEMS
Alberto Carini, Giovanni L. Sicuranza,                                                                    V.John Mathews
D.E.E.I.                                                  Department of Electrical Engineering
a
Universit di Trieste                                                   University of Utah
Via Valerio 10                                                    Salt Lake City UT 84112
34127 Trieste Italy                                                           USA
carini,sicuranz@ipl.univ.trieste.it                                                               mathews@ee.utah.edu

ABSTRACT                                     and in [6] they are used in order to derive more sim-
This paper presents two theorems for the exact in-                          ple and computationally ecient expressions for the
version and the th order inversion of a wide class of                       inverse system. However, because of the presence of
p

causal, discrete-time, nonlinear systems. The nonlinear                     higher order components, the denition of the th or-                 p

systems we consider are described by the input-output                      der inverse in [6] does not result in a unique inverse
    
relationship ( ) = ( ) + ( 1) ( 1) ,
                      system. Both the approaches of [6] and [7] lead to the
y n         g x n          f x n

where [] and [ ] are causal, discrete-time and non-
;y n
same result for the existence and the stability of the
g             f   ;

linear operators and the inverse function 1 [] exists.                     p th order inverse. If the linear part (i.e. the rst order
The exact inverse of such systems is given by ( ) =
g
Volterra kernel) of the system admit a Bounded In- H

1
( )
                  
( 1) ( 1) Similarly, the th or-
z n
put Bounded Output stable inverse, then the th order            p
g       u n        f z n         ;u n
h
:

p
inverse exists, it is BIBO stable and it depends only
der inverse is given by ( ) = p 1 ( )
z n        g  (    u n     f z n        from the rst Volterra operators of . In this paper,
p                                   H
i            
1) ( 1) where p 1  is the th order inverse                                 we accept the instability or the input dependent stabil-
;u n
 
g                  p
ity of the resulting system in order to obtain the exact
of  .
g                                                                       inverse of a particular class of discrete-time causal non-
linear systems. The systems we are interested in are
1. INTRODUCTION                                      described by the following input-output relationship:
                                    
Inversion of nonlinear systems is not a trivial task in                                 ( )=
y n           g x n    ( ) +   f x n(       1) (
;y n   1)        (1)
most situations. Not all nonlinear systems possess an                       where [] and [ ] are discrete-time causal nonlinear
inverse and many nonlinear systems admit an inverse                                g             f        ;

operators. Our main contribution is the derivation of
only for a certain subset of input signals. For these                       an expression for the exact inverse of the class of sys-
reasons, Schetzen has developed the theory of the th                    p
tems described by (1). The exact inverse of the system
order inverse of a nonlinear system whose input-output                      in (1) may not exist or may not be stable for certain in-
relation can be represented using Volterra series ex-                       put signals. However, even if the exact inverse cannot
pansions [7], [8]. The th order inverse of a nonlinear
p
be trivially derived, a more ecient realization of the
system is dened as the th order system which,
H                          p
th order inverse may be obtained. The ecient real-
connected in cascade with , results in a system whose
H
p

ization we propose is derived by the use of a nonlinear
Volterra kernels from the second up to the th order are  p
feedback.
zero. A th order system is one in which all the Volterra
p
The rest of this paper is organized as follows. The
kernels of order greater than are zero. The denition
p
inverse of the system in (1) is introduced in Section 2.
of the th order inverse was relaxed in [6] by allowing
p
An ecient th order inverse is derived in Section 3.
the inverse system to possess non-zero Volterra opera-                                       p

Section 4 presents some experimental results that con-
tors of order greater than . These operators do not af-
p
rm the usefulness of these inversion theorems. Con-
fect the rst Volterra operators of the cascade system
p
cluding remarks regarding the stability of the lters
This Work was partially supported by NATO Grant CRG.                    that results from the inversion procedure are discussed
950379 and Esprit LTR Project 20229 Noblesse.                               in Section 5.
2. THE INVERSE OF CERTAIN                                                             is the bilinear system
NONLINEAR SYSTEMS                                                                                                             N 1                                            N 1
X                                              X
In all our discussions we assume causal signals, i.e., all                                                  ( )= ( )
z n           u n                        i (
b u n                    )
i                       i (
a z n               i   )+
the signals are identically zero for time indices less than                                                                                i=1                                           i=1
N 1N 1
zero. The following theorem show how to evaluate the                                                                                               X X
ij z (n               ) (                    )
exact inverse of the system in (1).
c                  i u n                  j :
i=1 j =1
(7)
Theorem 1 Let [] and [ ] be causal nonlinear dis-
g               f       ;

crete operators and let the inverse operator [] exist.                            1

Then, the system described by the input-output rela-
g
3. TH ORDER INVERSESp

tionship                                                                                       Since the inverse system [] of Theorem 1 may not        g
1

( )=        1
h
( )

(        1) (             1)
i
(2)   always exist or may not be easy to derive, we now con-
z n     g            u n         f z n                 ;u n
sider the existence of the th order inverses of the same p

class of systems as before.
is the exact inverse of the system in (1).
Theorem 2 Let [] and [ ] be causal discrete-time
g                    f       ;
Proof: We demonstrate rst that the system in (2) is                                           nonlinear operators with convergent Volterra series ex-
the post-inverse of (1), i.e., a cascade interconnection of                                    pansion with respect to all the arguments. Moreover,
the system in (1) followed by the system in (2) results                                        let the pth order inverse gp 1 [] of the system g[] exist.
in an identity system. We proceed by mathematical                                              Then a pth order inverse of the causal discrete-time
induction. Let ( ) and ( ) represents the input and
x n                 y n
nonlinear system described in (1) is given by the fol-
output signals, respectively, of the system in (1). To                                         lowing input-output relationship
prove the theorem using induction, we assume that                                                                                      h                                                                     i
(
z n       i   )= (      x n           i)       8i >     0   :             (3)                        ( )= p1 ( )
z n       g            u n               f z n       (               1) (
;u n             1)          :            (8)
We must now show using (3) that                                                                Proof: As was the case for Theorem 1, we rst show
that the system in (8) is the th order post-inverse of                   p

z n ( )= ( )      x n                                     (4)   the system in (1). Using the same variables as in the
derivation of Theorem 1, we express ( ) as                                                    z n

when ( ) = ( ) for  . Now,
u k       y k                k      n                                                                                           h                                                                      i
z n ( ) =             gp
1
y n ( )          f z n       (               1) ( ;y n           1)
                                                                                                         h 
( ) =
z n           g
1
( )
y n   ( 1) ( 1)f z n                 ;y n                                                  =       gp
1
g x n   ( ) +
               
f x n       (         1) (    ;y n            1) +

               
=       g
1
( ) + ( 1) ( 1) +
g x n                  f x n                ;y n                                                         
f z n    (        1) (    ;y n                    1)
i
:                                        (9)
            
( 1) ( 1)
f z n                ;y n                     :

(5)                                     We proceed by mathematical induction. We assume
By substituting ( ) = ( ) from (3) into (5), it
z n           i         x n          i                                    that, for any greater than zero, the output (
i                     )                                                             z n               i
follows in a straightforward manner that ( ) = ( ).                     z n             x n    diers from (       ) only by p (
x n    ), a term whose
i                                T       n         i
We can prove in a similar manner that the system in (2)                                        Volterra series expansion in ( ) contains only kernels               x n
is also the pre-inverse of the system in (1), i.e., a cas-                                     of order larger than , i.e.,                    p
cade interconnection of the system in (2) followed by
the system in (1) results in an identity system. This                                                      z n (         )= (
i        x n          i   ) + p(   T           n           i)             8i >       0:                  (10)
completes the proof.
We have to prove that the Volterra series expansion of
Example 1: The inverse of the bilinear system                                                   ( ) ( ) have zero kernels of order up to . Since
z n                 x n                                                                                            p

N 1                         N 1                                       f[ ] admits a convergent Volterra series expansion, we
;

( ) = ( )+
X
i ( i) +
X
bi y (n  i)+
have from (10) that the Volterra series expansion of the
                                        
dierence ( 1) ( 1)                    ( 1) ( 1)
y n        x n                  a x n

i=1                      i=1                                                                f x n                 ;y n                                 f z n                    ;y n
N 1N 1
X X                                                     contains only kernels of order greater than , i.e.,                                                        p
cij x(n    i)y (n   j)
1) = 0+ p0 ( )
                                                                                              
i=1 j =1                                               f x n       (        1) ( ;y n         1)           f z n    (           1) (    ;y n                                  T           n ;

(6)                                                                                                                 (11)
where the Volterra kernels of p0 ( ) up to order are           T       n                                 p         contains only the second through th order Volterra   p

zero. Substituting (11) in (9), we get                                                                             kernels.
h                                  i                                  The computational cost expressed in multiplications
( ) = p 1 [ ( )] + p0 ( )
z n           g           (12)
g x n                T    n     :                             for the evaluation of (17) is 2( 1) + (N 21)N + ( +
N                        N

2)( 1). The corresponding computational cost for di-
p
The th order inverse of the operator [] derived in [7]
p                                                                     g                                   rectly computing the th order inverse of (14) as in [6]

p
is given by a th order truncated Volterra series whose
p
is + 2 + N (N +1) ( 1) If the order is greater
kernels depend only on the rst kernels of the Volterra
N             N                      p       :               p
2
than two the computational advantage of using (17) be-
p

series expansion of []. The th order inverse derived
comes evident. Implementing (17) has computational
g                    p

in [6] may have Volterra kernels of order greater than .
cost of ( 2 + ) multiplications while the method
p

However, the inverse still has a Volterra series expan-                                                                          O N         pN

sion with nite order of nonlinearity, and it depends                                                              in [6] requires ( 2 ) multiplications. In general, if we
O N p

only on the rst kernels of the Volterra series ex-                                                                want to derive a th order inverse for a Volterra lter
p

of order , the methodology suggested by Theorem 2
p

pansion of []. Consequently, it immediately follows                                                                                 q

is more convenient when is greater than . On the
g

from (12) that                                                                                                                                                  p                   q

other hand, when          only the rst Volterra oper-
p < q                          p

( ) = ( ) + p( )
z n                (13)
x n           T       n                                       ators are signicant for the evaluation of the th order              p

inverse. In this situation, both methods of inversion
and that the system in (8) is the th order post-inverse               p                                            require almost the same number of multiplications for
of the system in (1). We can prove in a similar manner                                                             computing each output sample.
that it is also a pre-inverse of the system in (1).
Example 2: We wish to derive a th order inverse for                   p
4. AN EXPERIMENTAL RESULT
the second order Volterra lter given by the following
expression:                                                                                                        We consider the th order inversion of the second order
p

N 1
X                                   N 1N 1
X X
Volterra lter with input-output relationship
( )=
y n                         i (
a x n             i)+                          bij x(n            )(
i x n            )
j :
y (n)         =   x(n)       x(n      1)   0:125x(n   2)+
i=0                                 i=0 j =i                                                                           0:3125x(n
2
3) + x (n)   0:3x(n)x(n         1)+
(14)                            0:2x(n)x(n   2)    0:5x(n)x(n    3)+
Let                                                                                                                                           2
0:5x (n   1)   0:3x(n    1)x(n   2)+
2
         
N 1
X
0:6x(n   1)x(n    3)   0:6x (n
2
2)+

g x n   ( ) = o ( )+ ( )
a x n                 x n                      j (
b0 x n             j   )       (15)                            0:5x(n   2)x(n    3)   0:1x (n   3):
(19)
j =0
The th order inverse derived applying Theorem 2,
p

and                                                                                                                where p 1 [] is computed as in [6], is compared with the
g

N 1                            N 1N 1                                                  pth order inverse obtained by directly using the method

( 1) =
        X
i (            )+
X X
ij x(n i)x(n j ):                  in [6]. In Figure 1 the points identied with  refer to
the th order inverse of the Theorem 2, while the points
f x n                                a x n           i                         b

i=1                               i=1 j =i                                                   p

(16)                                                               indicated with + refer to the th order inverse of [6].
p

According to Theorem 2, a th order inverse of (14) is     p
The plots in Figure 1a compare the computational cost

in multiplications for dierent orders of the inversion. p
N 1
X                                                    The computational eciency of the th order inverse       p
z n ( ) =               p 1 u(n)
g                   ai z (n                             i  )+                     of Theorem 2 over the inverse suggested in [6] can be
i=1                                          3
(17)      clearly seen in this gure. Figures 1b and 1c displays
N 1N 1
X X                                                                                 the mean-squared error (MSE) between the input sig-
bij z (n    i)z (n                                  )5
i=1 j =i
j        :
nal of the system in (19) and the output of its th order             p

inverse when connected in cascade to the system. The
The th order inverse p 1 [] can be computed itera-                                                                input signal was white and Gaussian-distributed with
zero mean value. Figure 1b presents the MSE in the
p                                       g

tively as in [6] and is given by
                1           
reconstruction of the input for dierent values of the
p ( ) = 1 p p 1 ( )
g
1
u n
1
g   ( )     (18)
q    g           u n               u n              ;             inverse lter order when the standard deviation of
p

the input signal was 0.05. Figure 1c shows the mean-
where 1 1 [] is the inverse of the rst Volterra opera-
g                                                                                                     square error values for dierent standard deviations of
tor of [] (i.e., 0 1 in our case) and p[] is the trun-
g                 a                                               q                                   the input signal for a fth-order inverse system. All
cated Volterra series expansion of the system [] that                                               g             the results presented are time averages of 1,000 sam-
−4                                             20
100                                   10                                             10
−5
80                                    10
multiplications

−6
10
60

MSE

MSE
−7                                             0
10                                             10
40                                     −8
10
20                                     −9
10
−10
0                                    10                                                       −2                 −1
0       2      4      6   8               0   2          4        6      8               10                  10
order p                                   order p                                     standard deviation
(a)                                       (b)                                              (c)

Figure 1: Experimental Results.

ples of the ensemble averages computed over fty in-                              [4] J.Lee, V.J.Mathews, \A Stability Result for RLS
dependent experiments. Values of the standard devi-                                   Adaptive Bilinear Filters," IEEE Signal Processing
ations for which a corresponding MSE value is absent                                  Letters, Vol.1, No.12, Dec. 1994, pp.191-193.
correspond to instability situations. We can see that
our approach give the similar or better performances                              [5] E.Mumolo, A.Carini, \A Stability Condition for
as the method in [6] till instability arises in the inverse                           Adaptive Recursive Second-Order Polynomial Fil-
system. In such situations, the performance of the th                                 ters," Signal Processing, Vol.54, No.1, Oct. 1996,
pp.85-90.
p

order inverse of [6] are also unacceptable.
[6] A.Sarti and S.Pupolin, \Recursive Techniques for
5. CONCLUDING REMARKS                                         the Synthesis of a th -Order Inverse of a Volterra
p

System," European Trans. on Telecomm., Vol.3,
This paper presented two theorems for the exact inverse                               No.4 Jul.-Aug. 1992, pp.315-322.
and the th order inverse of a wide class of discrete-
p

time nonlinear systems. As in the linear case, even                               [7] M.Schetzen, \Theory of the th-Order Inverses of
p

if a nonlinear system is BIBO stable its inverse sys-                                 Nonlinear Systems," IEEE Trans. on Circuits and
tem may be unstable. The inverse systems we consider                                  Systems, Vol.CAS-23, No.5, May 1976, pp.285-291.
in this paper are in most cases recursive nonlinear l-
ters and therefore may possess poor stability proper-                             [8] M.Schetzen, The Volterra and Wiener Theories of
ties. Consequently, the stability of such systems must                                Nonlinear Systems, John Wiley and Sons, Inc. New
be tested after the inversion of the lter. Stability of                              York, NY, 1980.
recursive nonlinear systems is still a topic of active re-                        [9] T.Siu, M.Schetzen, \Convergence of Volterra series
search. Some useful results for the stability of recursive                            representation and BIBO stability of Bilinear Sys-
polynomial lters can be found in [1, 2, 3, 4, 5, 9].                                 tems," Int. J. System Science, Vol.22, No.12, Dec.
1991, pp.2679-2684.
6. REFERENCES
[1] T.Bose, M.Q.Chen, \BIBO Stability of the discrete
bilinear systems," Digital Signal Processing: a Re-
view Journal, Vol.5, No.3, July 1995, p.p.160-166.
[2] St. Kotsios, N.Kalouptsidis, \BIBO Stability Cri-
teria for a Certain Class of Discrete Nonlinear Sys-
tems," Int. J. Control, vol.58, No.3, Sep. 1993,
pp.707-730.
[3] J.Lee, V.J.Mathews, \A Stability Condition for
Certain Bilinear Systems," IEEE Trans. on Signal
Processing, Vol.42, No.7, July 1994, pp.1871-1873.

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